Solving Inequalities

Start by finding the roots of the equation by changing the inequality to an equal sign.

Now, make a number line with the two roots:

Pick a number less than and plug it into the inequality to see if it holds.

For ,

is clearly not true. The solution set cannot be .

Next, pick a number between .

For ,

is true so the solution set must include .

Finally, pick a number greater than .

For ,

is clearly not the so the solution set cannot be .

Thus, the solution set for this inequality is .

Add 26 on both sides.

Divide by two on both sides.

The answer is:

Subtract nine from both sides.

Divide by negative 3 on both sides. We will need to switch the sign.

The answer is:

First, we distribute the and then collect terms:

Now we solve for x, taking care to change the direction of the inequality if we divide by a negative number:

Add six on both sides.

Divide by three on both sides.

The answer is:

First, find the zeroes of the numerator and the denominator. This will give the boundary points of the intervals to be tested.

;

Since the numerator may be equal to 0, and are included as solutions. However, since the denominator may not be equal to 0, is excluded as a solution.

Now, test each of four intervals for inclusion in the solution set by substituting one test value from each:

Let's test :

This is false, so is excluded from the solution set.

Let's test :

This is true, so is included in the solution set.

Let's test :

This is false, so is excluded from the solution set.

Let's test :

This is true, so is included in the solution set.

The solution set is therefore .

It can be tricky because there's a negative sign in the equation, but we never end up multiplying or dividing by a negative, so there's no need to change the direction of the inequality. We simply divide by and multiply by :

The first thing we can do is clean up the right side of the equation by distributing the , and combining terms:

Now we can combine further. At some point, we'll have to divide by a negative number, which will change the direction of the inequality.

First, we distribute the through the equation:

Now, we collect and combine terms:

In order to solve this inequality, we need to apply each mathematical operation to all three sides of the equation. Let's start by subtracting from all the sides:

Now we divide each side by . Remember, because the isn't negative, we don't have to flip the sign: